The aim of this article is to give Sturm-Picone type theorems for the pair of second order nonlinear elliptic differential equations div(p1(x)|∇u|∇u) + q1(x)f1(u) + r1(x)g1(u) = 0, div(p2(x)|∇v|∇v) + q2(x)f2(v) + r2(x)g2(v) = 0, where | · | denotes the Euclidean length and ∇ = ( ∂ ∂x1 , . . . , ∂ ∂xn )T (the superscript T denotes the transpose). Our results include some earlier results and gene...

An identity of the Picone type for fourth-order half-linear ordinary differential operators of the form lα[x] ≡ (pφ(x′′))′′ − (rφ(x′))′ + qφ(x) and Lα[y] ≡ (Pφ(y′′))′′ − (Rφ(y′))′ +Qφ(y). where φ(u) := |u|α−1u, α > 0, u ∈ R, and p, q, r, P,Q and R are continuous functions on a given interval I is derived and then Sturmian comparison theory for the corresponding fourth-order equations lα[x] = 0 ...

We consider the oscillatory property of the following p(t)-Laplacian equations −(|u′|p(t)−2u′)′ = 1/tθ(t)g(t,u), t > 0. Since there is no Picone-type identity for p(t)Laplacian equations, it is an unsolved problem that whether the Sturmian comparison theorems for p(x)-Laplacian equations are valid or not. We obtain sufficient conditions of the oscillatory of solutions for p(t)-Laplacian equations.

As the impulsive differential equations are useful in modelling many real processes observed in physics, chemistry, biology, engineering, etc., see [1, 11, 13, 20, 21, 22, 25, 26, 27], there has been an increasing interest in studying such equations from the point of view of stability, asymptotic behavior, existence of periodic solutions, and oscillation of solutions. The classical theory can b...

Recently, an identity of the Picone type for the weighted p-biharmonic operator was established by the present author [5] and employed in proving comparison theorems and other qualitative results for a pair of fourth-order elliptic partial differential equations of the form ∆ ( a(x)|∆u|p−2∆u ) − c(x)|u|p−2u = 0, (1.1) and ∆ ( A(x)|∆v|p−2∆v ) − C(x)|v|p−2v = 0, (1.2) where p > 1, a,A ∈ C(Ḡ,R+), ...

Abstract The aim of this paper is to study the oscillation solutions nonlinear degenerate elliptic equation in Heisenberg group $H^{n}$ H n . We first derive a critical inequality Based on it, we establish Picone-type differential and Sturm-type comparison principle. Th...

Two identities of the Picone type for a class of half-linear differential systems in the plane are established and the Sturmian comparison theory for such systems is developed with the help of these new formulas.